Resolving singularities of varieties and families

Dan Abramovich Brown University Joint work with Michael Temkin and Jaroslaw Wlodarczyk

July 2018

Abramovich Resolving singularities of varieties and families July 2018 1 / 25 On singularities

figrures by Herwig Hauser, https://imaginary.org/gallery/herwig-hauser-classic

Singularities are beautiful. Yet we get rid of them.

Abramovich Resolving singularities of varieties and families July 2018 2 / 25 Resolution of singularities

Definition A resolution of singularities X 0 → X is a modificationa with X 0 nonsingular inducing an isomorphism over the smooth locus of X .

aproper birational map

Theorem (Hironaka 1964) A variety X over a field of characteristic 0 admits a resolution of singularities X 0 → X , so that the exceptional locus E ⊂ X 0 is a simple normal crossings divisor.a

aCodimension 1, smooth components meeting transversally

Always characteristic 0 . . .

Abramovich Resolving singularities of varieties and families July 2018 3 / 25 Resolution of families: dim B = 1 Question When are the singularities of a morphism X → B simple?

Q ai If dim B = 1 the simplest one can have by modifying X is t = xi , Q and if one also allows base change, can have t = xi . [Kempf–Knudsen–Mumford–Saint-Donat 1973]

Question What makes these special?

Abramovich Resolving singularities of varieties and families July 2018 4 / 25 Log smooth schemes and log smooth morphisms

∗ n A toric variety is a normal variety on which T = (C ) acts algebraically with a dense free orbit. Zariski locally defined by equations between monomials.

A variety X with divisor D is toroidal or log smooth if ´etalelocally it looks like a toric variety Xσ with its toric divisor Xσ r T . Etale´ locally it is defined by equations between monomials.

A morphism X → Y is toroidal or log smooth if ´etalelocally it looks like a torus equivariant morphism of toric varieties. The inverse image of a monomial is a monomial.

Abramovich Resolving singularities of varieties and families July 2018 5 / 25 Resolution of families: higher dimensional base

Question When are the singularities of a morphism X → B simple?

The best one can hope for, after base change, is a semistable morphism: Definition (ℵ-Karu 2000) A log smooth morphism, with B smooth, is semistable if locally

t1 = x1 ··· xl1 . . . . t = x ··· x m lm−1+1 m

In particular log smooth. Similar definition by Berkovich, all following de Jong.

Abramovich Resolving singularities of varieties and families July 2018 6 / 25 The semistable reduction problem

Conjecture [ℵ-Karu] Let X → B be a dominant morphism of varieties.

(Loose) There is an alteration B1 → B and a modification X1 → (X ×B B1)main such that X1 → B1 is semistable.

(Tight) If the geometric generic fiber Xη¯ is smooth, such X1 → B1 can be found with Xη¯ unchanged.

One wants the tight version in order to compactify smooth families. I’ll describe progress towards that. Major early results by [KKMS 1973], [de Jong 1997]. Wonderful results in positive and mixed characteristics by de Jong, Gabber, Illusie and Temkin.

Abramovich Resolving singularities of varieties and families July 2018 7 / 25 Toroidalization and weak semistable reduction Back to characteristic 0 Theorem (Toroidalization, ℵ-Karu 2000, ℵ-K-Denef 2013)

There is a modification B1 → B and a modification X1 → (X ×B B1)main such that X1 → B1 is log smooth and flat.

Theorem (Weak semistable reduction, ℵ-Karu 2000)

There is an alteration B1 → B and a modification X1 → (X ×B B1)main such that X1 → B1 is log smooth, flat, with reduced fibers.

Passing from weak semistable reduction to semistable reduction is a purely combinatorial problem [ℵ-Karu 2000], proven by [Karu 2000] for families of surfaces and threefolds, and whose restriction to rank-1 valuation rings is proven in a preprint by [Karim Adiprasito - Gaku Liu - Igor Pak - Michael Temkin].

Abramovich Resolving singularities of varieties and families July 2018 8 / 25 Applications of weak semistable reduction

(with a whole lot of more input) Theorem (Karu 2000; K-SB 97, Alexeev 94, BCHM 11) The moduli space of stable smoothable varieties is projective.

Theorem (Viehweg-Zuo 2004) The moduli space of canonically polarized manifolds is Brody hyperbolic.

Theorem (Fujino 2017) Nakayama’s numerical logarithmic Kodaira dimension is subadditive in families X → B with generic fiber F :

κσ(X , DX ) ≥ κσ(F , DF ) + κσ(B, DB ).

Abramovich Resolving singularities of varieties and families July 2018 9 / 25 Main result

The following result is work-in-progress. Main result (Functorial toroidalization, ℵ-Temkin-W lodarczyk) Let X → B be a dominant log morphism.

There are log modifications B1 → B and X1 → (X ×B B1)main such that X1 → B1 is log smooth and flat; this is compatible with log base change B0 → B; this is functorial, up to base change, with log smooth X 00 → X .

This implies the tight version of the results of semistable reduction type.

Abramovich Resolving singularities of varieties and families July 2018 10 / 25 In virtue of functoriality

Theorem (Temkin) Resolution of singularities holds for excellent schemes, complex spaces, nonarchimedean spaces, p-adic spaces, formal spaces and for stacks.

This is a consequence of resolution for varieties and schemes, functorial for smooth morphisms (submersions). Moreover Wlodarczyk showed that if one seriously looks for a resolution functor, one is led to a resolution theorem. Our main result will lead to this generality on families. Current application of our main result: Theorem (Deng 2018) The moduli space of minimal complex projective manifolds of general type is Kobayashi hyperbolic.

Abramovich Resolving singularities of varieties and families July 2018 11 / 25 dim B = 0: log resolution via principalization To resolve log singularities, one embeds X in a log smooth Y ...... which can be done locally. One reduces to principalization of IX (Hironaka, Villamayor, Bierstone–Milman). Theorem (Principalization . . . ℵ-T-W) Let I be an ideal on a log smooth Y . There is a functorial logarithmic 0 0 morphism Y → Y , with Y logarithmically smooth, and IOY 0 an invertible monomial ideal.

2 2 2 Figure: The ideal (u , x ) and the result of blowing up the origin, IE . Here u is a monomial but x is not. Abramovich Resolving singularities of varieties and families July 2018 12 / 25 Logarithmic order

Principalization is done by order reduction, using logarithmic derivatives. ∂ for a monomial u we use u ∂u . ∂ for other variables x use ∂x . Definition Write D≤a for the sheaf of logarithmic differential operators of order ≤ a. The logarithmic order of an ideal I is the minimum a such that D≤aI = (1).

Take u, v monomials, x free variable, p the origin. 2 ∂ logordp(u , x) = 1 (since ∂x x = 1) 2 2 2 logordp(u , x ) = 2 logordp(v, x ) = 2 ≤1 ≤2 logordp(v + u) = ∞ since D I = D I = ··· = (u, v).

Abramovich Resolving singularities of varieties and families July 2018 13 / 25 The monomial part of an ideal

Definition M(I) is the minimal monomial ideal containing I.

Proposition (Koll´ar, ℵ-T-W) (1) In cahracteristic 0, M(I) = D∞(I). In particular maxp logordp(I) = ∞ if and only if M(I) 6= 1. (2) Let Y0 → Y be the normalized blowup of M(I). Then

M := M(I)OY0 = M(IOY0 ), and it is an invertible monomial ideal,

and so IOY0 = I0 ·M with maxp logordp(I0) < ∞.

(1)⇒(2)

DY0 is the pullback of DY , so (2) follows from (1) since the ideals have the same generators.

Abramovich Resolving singularities of varieties and families July 2018 14 / 25 The monomial part of an ideal - proof

Proof of (1), basic affine case.

Let OY = C[x1,..., xn, u1,..., um] and assume M = D(M). The operators ∂ ∂ 1, u1 ,..., ul ∂u1 ∂ul commute and have distinct systems of eigenvalues on the eigenspaces u C[x1,..., xn], for distinct monomials u. Therefore M = ⊕uMu with ideals Mu ⊂ C[x1,..., xn] stable under derivatives,

so each Mu is either (0) or (1). In other words, M is monomial. ♠

The general case requires more commutative algebra.

Abramovich Resolving singularities of varieties and families July 2018 15 / 25 dim B = 0: sketch of argument

≤a−1 In cahracteristic 0, if logordp(I) = a < ∞, then D I contains an element x with derivative 1, a maximal contact element. Carefully applying induction on dimension to an ideal on {x = 0} gives order reduction (Encinas–Villamayor, Bierstone–Milman, Wlodarczyk): Proposition (. . . ℵ-T-W) Let I be an ideal on a logarithmically smooth Y with

max logord (I) = a. p p

There is a functorial logarithmic morphism Y1 → Y , with Y1 logarithmically smooth, such that IOY 0 = M·I1 with M an invertible monomial ideal and max logord (I1) < a. p p

Abramovich Resolving singularities of varieties and families July 2018 16 / 25 Arbitrary B

(Work in progress) Main result (ℵ-T-W) Let Y → B a logarithmically smooth morphism of logarithmically smooth 0 schemes, I ⊂ OY an ideal. There is a log morphism B → B and functorial log morphism Y 0 → Y , with Y 0 → B0 logarithmically smooth, and IOY 0 an invertible monomial ideal.

This is done by relative order reduction, using relative logarithmic derivatives. Definition ≤a Write DY /B for the sheaf of relative logarithmic differential operators of order ≤ a. The relative logarithmic order of an ideal I is the minimum a ≤a such that DY /B I = (1).

Abramovich Resolving singularities of varieties and families July 2018 17 / 25 The new step

∞ M := DY /B I is an ideal which is monomial along the fibers. ∞ relordp(I) = ∞ if and only if M := DY /B I is a nonunit ideal. Monomialization Theorem [ℵ-T-W] Let Y → B a logarithmically smooth morphism of logarithmically smooth schemes, M ⊂ OY an ideal with DY /B M = M. There is a log morphism 0 0 0 B → B with saturated pullback Y → B , such that MOY 0 a monomial ideal. After this one can proceed as in the case “dim B = 0”.

Abramovich Resolving singularities of varieties and families July 2018 18 / 25 Proof of Monomialization Theorem, special case

Let Y = Spec C[u, v] → B = Spec C[w] with w = uv, and M = (f ). Proof in this special case. Every monomial is either uαw k or v αw k . ∂ ∂ Once again the operators 1, u ∂u − v ∂v commute and have different eigenvalues on uα, v α. P α P β Expanding f = u fα + v fβ, the condition M = DY /B M gives that only one term survives, α say f = u fα, with fα ∈ C[w]. Blowing up (fα) on B has the effect of making it monomial, so f becomes monomial. ♠

The general case is surprisingly subtle.

Abramovich Resolving singularities of varieties and families July 2018 19 / 25 Order reduction: Example 1

Consider Y1 = Spec C[u, x] and D = {u = 0}. Let I = (u2, x2). If one blows up (u, x) the ideal is principalized:

0 0 2 I on the u-chart Spec [u, x ] with x = x u we have IO 0 =( u ), C Y1 0 0 0 2 I on the x-chart Spec C[u , x] with u = xu we have IOY 0 =( x ), I which is exceptional hence monomial. This is in fact the only functorial admissible blowing up.

Abramovich Resolving singularities of varieties and families July 2018 20 / 25 Order reduction: Example 2

Consider Y2 = Spec C[v, x] and D = {v = 0}. Let I = (v, x2). Example 1 is the pullback of this via the log smooth v = u2. Functoriality says: we need to blow up an ideal whose pullback is (u, x). This means we need to blow up( v 1/2, x). What is this? What is its blowup?

Abramovich Resolving singularities of varieties and families July 2018 21 / 25 Kummer ideals

Definition A Kummer monomial is a monomial in the Kummer-´etaletopology of Y (like v 1/2). A Kummer monomial ideal is a monomial ideal in the Kummer-´etale topology of Y . A Kummer center is the sum of a Kummer monomial ideal and the ideal of a log smooth subscheme. 1/d 1/d Locally (x1,..., xk , u1 ,... u` ).

Abramovich Resolving singularities of varieties and families July 2018 22 / 25 Blowing up Kummer centers

Proposition Let J be a Kummer center on a logarithmically smooth Y . There is a universal proper birational Y 0 → Y such that Y 0 is logarithmically smooth and JOY 0 is an invertible ideal.

Example 0 Y = Spec C[v], with toroidal structure associated to D = {v = 0}, and J = (v 1/2).

There is no log scheme Y 0 satisfying the proposition. √ There is a stack Y 0 = Y ( D), the Cadman–Vistoli root stack, satisfying the proposition!

Abramovich Resolving singularities of varieties and families July 2018 23 / 25 Example 2 concluded

Consider Y2 = Spec C[v, x] and D = {v = 0}. Let I = (v, x2) and J = (v 1/2, x). 0 associated blowing up Y → Y2 with charts: 0 0 0 2 0 2 I Yx := Spec C[v, x, v ]/(v x = v), where v = v/x (nonsingular scheme).

F Exceptional x = 0, now monomial. 2 2 F I = (v, x ) transformed into (x ), invertible monomial ideal. 1/2 F Kummer ideal (v , x) transformed into monomial ideal( x). 1/2 I The v -chart: 0    F stack quotient Xv 1/2 := Spec C[w, y] µ2 , F where y = x/w and µ2 = {±1} acts via (w, y) 7→ (−w, −y). F Exceptional w = 0 (monomial). 2 2 F (v, x ) transformed into invertible monomial ideal (v) = (w ). 1/2 F (v , x) transformed into invertible monomial ideal( w).

Abramovich Resolving singularities of varieties and families July 2018 24 / 25 Proof of proposition

Let J be a Kummer center on a logarithmically smooth Y . There is a universal proper birational Y 0 → Y such that Y 0 is a logarithmically smooth stack and JOY 0 is an invertible ideal.

˜ ˜ Choose a stack Y with coarse moduli space Y such that J := JOY˜ is an ideal. Let Y˜ 0 → Y˜ be the blowup of J˜, with exceptional E. ˜ 0 Let Y → BGm be the classifying morphism of IE . 0 ˜ 0 Y is the relative coarse moduli space of Y → Y × BGm. One shows this is independent of choices. ♠

Abramovich Resolving singularities of varieties and families July 2018 25 / 25